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User talk:Vel!/Prime blocks
This is different from how I defined prime blocks - in my version successor ordinals had the empty set as a prime block, so for example P(omega*2) = union_{n < p} {omega + n}, it did not include {n | n < p}. But your version should work just fine. Deedlit11 (talk) 02:43, March 13, 2013 (UTC) :I'm starting from scratch in hopes of duplicating BEAF as exactly as possible. FB100Z • talk • 03:19, March 13, 2013 (UTC) Nice, that looks like it works! However, it's not even necessary to define \(\gamma\); just define \(\Pi_p(\alpha) = \Pi_p(\alphap) \) and that works as well! I don't think \(B_{\alpha}(\omega) \) is well-defined, since \(\Pi_{\omega}(\alpha) = \Pi_{\omega}(\alpha\omega) = \Pi_{\omega}(\alpha) \) using either definition, although I guess the obvious choice for \(\Pi_{\omega}(\alpha)\) would be \(\alpha\) itself. Deedlit11 (talk) 21:34, March 13, 2013 (UTC) When I was mentioning prime blocks of ordinals under some blog post this is EXACTLY what I meant, i.e. predecessors and fragment of fundamental sequence (it is provable, I think, that \(\beta +\gamma p\) is equivalent to fundamental sequence of ordinal) LittlePeng9 (talk) 20:55, March 13, 2013 (UTC) :D'oh! I must've thought that \((\omega \times 2)n = n\) instead of \(\omega + n\). That makes the definition much simpler. :Anyways, now that I appear to have prime blocks down, the next step is to define BEAF itself! FB100Z • talk • 21:11, March 13, 2013 (UTC) @LittlePeng9, I understood you; my version of the prime blocks is basically what you said in your post. It was just an issue of getting the specifics right. @FB100Z: You've rediscovered the slow-growing hierarchy! I think I'll make a short blog post about it. Deedlit11 (talk) 21:34, March 13, 2013 (UTC) :Yeah, the B function reminded me a lot about SGH. :So, how does this new definition look? FB100Z • talk • 21:47, March 13, 2013 (UTC) :Unfortunately, \(E_{\gamma}(\alpha)\) does not correctly find the \(\gamma\)th coefficient of \(\alpha\); it only works for the last term in the sum. Deedlit11 (talk) 22:39, March 13, 2013 (UTC) :There's still a problem: it doesn't work if \(\alpha < \omega^{\gamma+1}\), since \(\beta_1\) is required to be at least \(\omega^{\gamma+1}\). Deedlit11 (talk) 23:13, March 13, 2013 (UTC) :Here's a fix: :\= \max\{n \in \mathbb{N}_0|\exists \beta_2 < \omega^\gamma, \beta_1: \omega^{\gamma+1} \beta_1 + \omega^\gamma \times n + \beta_2 = \alpha\}\ :Deedlit11 (talk) 23:24, March 13, 2013 (UTC) :Okay, it looks good now. I have to wonder, though, at the rationale behind going from a single ordinal to multiple ordinals to make it more like BEAF, and then coding up multiple ordinals into a single ordinal. I think the definition is simpler if we just leave it with multiple ordinals. Anyway, good work. Deedlit11 (talk) 23:36, March 13, 2013 (UTC) Evaluation of \(B_\alpha (p)\) I'm getting a more complicated result for \(B_{\epsilon_1}(p)\): \(\epsilon_10 = \epsilon_0 + 1 \rightarrow B_{\epsilon_1}(0) = B_{\epsilon_0 + 1}(0) = B_{\epsilon_0}(0) + 1 = 1+1 = 2 \) \(B_{\epsilon_1}(1) = B_{\omega^{\epsilon_0 + 1}}(1) = B_{\omega^{\epsilon_0} * 1} (1) = B_{\epsilon_0}(1) = B_{\omega}(1) = B_{1}(1) = B_0(2) = 2 \) \(B_{\epsilon_1}(2) = B_{\omega^{\omega^{\epsilon_0 + 1}}}(2) = B_{\omega^{\epsilon_0 + \epsilon_0}}(2) = B_{\omega^{\epsilon_0 + \omega^{\omega}}}(2) = B_{\omega^{\epsilon_0 + \omega^2}}(2) = B_{\omega^{\epsilon_0 + \omega * 2}}(2)\) \(= B_{\omega^{\epsilon_0 + \omega + 2}}(2) = B_{\omega^{\epsilon_0 + \omega + 1} * 2}(2) = 2 B_{\omega^{\epsilon_0 + \omega + 1}}(2) = 2 B_{ \omega^{\epsilon_0 + \omega} * 2}(2) = 4 B_{ \omega^{\epsilon_0 + \omega}}(2) \) \(= 4 B_{ \omega^{\epsilon_0 + 2}}(2) = 8 B_{ \omega^{\epsilon_0 + 1}}(2) = 16 B_{ \omega^{\epsilon_0}}(2) = 16 B_{\epsilon_0}(2) = 16 B_{\omega^{\omega}}(2) = 16(2^2) = 64\) I don't feel up to evaluating exactly the higher values, but for example \(\epsilon_1 3 = \omega^{\omega^{\omega^{\epsilon_0 + 1}}} 3 = \omega^{\omega^{\epsilon_0 * 3}} 3 = \omega^{\omega^{\epsilon_0 *2 + \omega^{\omega^{\omega}}}} 3 \) where n means we keep taking the nth element of the fundamental sequence until we get to a successor ordinal or 0. So for n = 3 we get a power tower of height 5; for general n we get a power tower of height 2n-1, so I'm guessing that \(B_{\epsilon_1}(n) \approx ^{2n-1}n \). Deedlit11 (talk) 04:49, March 14, 2013 (UTC) :I guess my intuition failed me :P FB100Z • talk • 15:54, March 14, 2013 (UTC) Holy grail Saibian mentioned that the part of holy grail of googologists is formalizing BEAF in the article about Cascading-E notation. Ikosarakt1 (talk ^ ) 11:41, August 15, 2013 (UTC) I think this isn't what Saibian imagined, as BEAF doesn't use notion of ordinals. & operator would be very non-trivial mapping \(\text{Ord}\times\mathbb{N}\rightarrow\text{Ord}\) in this notation, which I don't believe anyone constructed yet. LittlePeng9 (talk) 12:22, August 15, 2013 (UTC) But this is why it is so interesting, as it hypothethically can take arbitrary structure and construct an array based on its definition. It means that SCG(X) & n, TREE(X) & n and BB(X) & n may be possible. One of the signs of that would be the SCG(n) entries when SCG(X) & n, being constructed. Sometimes I'm trying to write down the complete set of rules leading up to pentational arrays. The best thing what I found for getting past tetrational arrays is usage of => signs which separate one power tower from other and with A => B: A indicates how many X's closed under exponentiation. The ruleset for this would be clunky compared to the one needed to get to the same strength in FGH. Ikosarakt1 (talk ^ ) 12:50, August 15, 2013 (UTC)